Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Hurwitz quaternions are defined as:

$$H_u = \left\{a+bi+cj+dk\mid (a,b,c,d) \in\mathbb{Z}\mbox{ or }(a,b,c,d) \in\mathbb{Z}+\frac{1}{2}\right\}$$ (that is, all integer or half integer quaternions).

I'm trying to show that if $a$ is a unit, then $||a||=1$ (the usual quaternion norm), but I can only sort of show that using very tedious calculations. Is there a short(er) and more elegant way to see this, preferably purely algebraic?

share|improve this question
add comment

1 Answer

up vote 7 down vote accepted

Suppose $x$ is a unit. Then $x^{-1}$ satisfies $\lVert x \rVert \lVert x^{-1} \rVert = 1$. So $\lVert x^{-1} \rVert = \lVert x \rVert ^{-1}$, and in particular one of $\lVert x \rVert$, $\lVert x^{-1} \rVert$ must be less than or equal to 1. But it's not hard to see that in fact the norm is an integer and not zero, so it must be 1.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.